Unbalanced One-way Model Research Articles

Calculations of level probabilities for normal random variables with unequal variances with applications to Bartholomew's test in unbalanced one-way models

An effective procedure is presented to compute the level probabilities under simple order of independent normal random variables with unequal variances. These level probabilities have an application in the unbalanced one-way layout for comparing k treatment effects where Bartholomew (1959. Biometrika 48, 325–332, 1961a. J. Roy. Statist. Soc. Ser. B 23 239–281, 1961b. Biometrika 48, 325–332) proposed the likelihood ratio test for testing the homogeneity of the treatment effects against the simply ordered alternative hypothesis. Although there is some literature showing that Bartholomew's test has good properties, its null distribution for the unbalanced model has been difficult to calculate except for small k. The calculation of the level probabilities allows the computation of the p-values of Bartholomew's test for unbalanced models.

Calculations of level probabilities in unbalanced one-way models with applications to Bartholomew's test(第13回日本計算機統計学会大会報告)

Calculations of level probabilities in unbalanced one-way models with applications to Bartholomew's test(第13回日本計算機統計学会大会報告)

Choice of noninformative priors for the variance components of an unbalanced one-way random model

Several noninformative priors for the variance components of an unbalanced one-way model are proposed. A simulation study is performed to compare the posterior means of the variance components based on the different priors. These estimators are also compared to the restricted maximum likelihood estimator (REML). Under the squared error loss, this simulation study indicates that the invariant priors lead to the optimal estimators of the variance components.

Graphical Representations of Tukey's Multiple Comparison Method

Abstract Tukey's multiple comparison method is widely implemented in statistical packages. We list the inferential and perceptual tasks a graphical representation of Tukey's method may be required to perform, and show that the most commonly used graphical representations (underlining, line-by-line plotting, notched boxplots, and comparison circles) are all lacking in some respect. A new graphical representation, based on the mean-mean scatterplot, is proposed instead. This representation performs all the inferential and perceptual tasks not only in an unbalanced one-way model, but also in a general linear model. An event-driven, color-graphics implementation is presented. Finally, issues for future research, such as color perception and comparative experimentation, are briefly discussed.

A Comparison of Variance Component Estimates for Arbitrary Underlying Distributions

Abstract Large-sample covariance matrices for the analysis of variance (ANOVA), minimum norm quadratic unbiased estimator (MINQUE), restricted maximum likelihood (REML), and maximum likelihood (ML) estimates of variance components are presented for the unbalanced one-way model when the underlying distributions are not necessarily normal. The limiting variances depend on the design sequence, on the actual values of the variance components, and on the kurtosis parameters of the underlying distributions. (The skewness parameters and other moments do not affect the limiting distributions.) Because all estimates are consistent and asymptotically normal, it is reasonable to compare the estimates using their asymptotic variances. Thus the efficiency of one estimate relative to another is defined as the ratio of their asymptotic variances, and these efficiencies are evaluated numerically and analytically for a variety of nonnormal situations. Various authors, including Hocking and Kutner (1975), Corbeil and Searl...

The Existence of Asymptotically Unbiased Nonnegative Quadratic Estimates of Variance Components in ANOVA Models

Abstract Conditions for the existence of asymptotically unbiased, nonnegative quadratic estimates of individual variance components in general ANOVA models are investigated. A simple necessary condition for general models is obtained. Necessary and sufficient conditions are obtained in the case of random, balanced nested classification models and the random unbalanced one-way model. The investigation demonstrates the need for exercising caution when employing nonnegative quadratic estimates of individual variance components in practice.